If the patient chose to ride a taxi instead of taking the time to wait for an ambulance, then that is his choice. Many farmers and other industrial workers may have to drive, walk or crawl great distances to get help of any type if they suffer an injury.
However, I believe it is irresponsible in the U.S. for a Paramedic to tell a patient to take a taxi after a traumatic amputation with or without the amputated part. One never knows how a person will react to their injury once the initial shock wears off. While it may just be a finger, to some that could be a career ender. Even for those in EMS, if it is your dominant hand, you may have to relearn many skills and it may be difficult for you to do your job to where you may have to find another career path. Realizing that can be devastating if one is alone in a taxi without support of any type and can be dangerous if the taxi driver does not know how to react to your emotions. Bleeding can also become an issue again as the spasms may relax. At least with a Paramedic, pain and bleeding control can hopefully be done as well as some support for the emotional issues surrounding these injuries.
If a hospital has properly cleaned stump and determined that another facility is needed to provide more care, few hospitals are going to release a patient to go by taxi (in the U.S.) especially after they have initiated an IV and have given antibiotics and pain medicine. The wound may still be open and at risk for infection or improper healing. They would also probably have to remove the IV catheter and the patient would have to have it restarted at another facility. Yes, there are some injuries that patients are sent home with a splint to wait for a surgical appointment but there are also many that spend their wait in a hospital to reduce risk and further damage until a specialist can be seen.